Quantifying Empirical QQ Plots: Stock Markets, Executive Pay, and Weather

Abstract

This article introduces two new ideas in the use of QQ plots as visual aids to explore the comparative shapes of distributions. First, we investigate the situation where both x and y distributions are empirical. We derived a procedure for the QQ plot that is based on geometric mean regression, and is simple enough to be calculated in a spreadsheet. We also indicate how this procedure may be robustified, and how it maps onto the one-empirical/one-theoretical QQ plot normally encountered. The second innovation is to use bootstrap sampling to guard against over-interpreting what is seen in the QQ plot. This may also be implemented on a spreadsheet. We illustrate the method with three worked examples that compare the distributions of: (i) UK versus Chinese executive pay, (ii) daily returns for the Shanghai versus the Shenzhen stock markets, and (iii) annual temperatures in the USA, 1895-1951 versus 1952-2007.